МАТЕМАТИКА Approximation of Continuous 2 π-Periodic Piecewise Smooth Functions by Discrete Fourier Sums

Let N be a natural number greater than 1. Select N uniformly distributed points tk = 2πk/N + u (0 6 k 6 N − 1), and denote by Ln,N (f) = Ln,N (f, x) (1 6 n 6 N/2) the trigonometric polynomial of order n possessing the least quadratic deviation from f with respect to the system {tk} k=0 . Select m + 1 points −π = a0 < a1 < . . . < am−1 < am = π, where m > 2, and denote Ω = {ai}mi=0. Denote by C Ω a class of 2π-periodic continuous functions f , where f is r-times differentiable on each segment ∆i = [ai, ai+1] and f (r) is absolutely continuous on ∆i. In the present article we consider the problem of approximation of functions f ∈ C2 Ω by the polynomials Ln,N (f, x). We show that instead of the estimate |f(x) − Ln,N (f, x)| 6 c lnn/n, which follows from the well-known Lebesgue inequality, we found an exact order estimate |f(x) − Ln,N (f, x)| 6 c/n (x ∈ R) which is uniform with respect to n (1 6 n 6 N/2). Moreover, we found a local estimate |f(x) − Ln,N (f, x)| 6 c(ε)/n2 (|x − ai| > ε) which is also uniform with respect to n (1 6 n 6 N/2). The proofs of these estimations are based on comparing of approximating properties of discrete and continuous finite Fourier series.


INTRODUCTION
We begin by establishing some notations.Let Ω be a set of m + 1 points {a i } m i=0 (m > 2) such that −π = a 0 < < a 1 < . . .< a m−1 < a m = π.We denote by C 0,r Ω the class of 2π-periodic functions with r absolutely continuous derivatives on each interval (a i , a i+1 ) and by C r Ω the subclass of all continuous functions in C 0,r Ω (here we say that a function f is absolutely continuous on an interval (a, b) if the function f is absolutely continuous on the segment [a, b], where f (x) = f (x) for x ∈ (a, b), f (a) = f (a + 0), and f (b) = f (b − 0)).
We denote by L n,N (f, x) a trigonometric polynomial of order n possessing the least quadratic deviation from the function f at the points {t j } N −1 i=0 , where t j = u + 2πj/N , n N/2, N 2, and u ∈ R. In other words, L n,N (f, x) provides the minimum for the sum N −1 j=0 |f (t j ) − T n (t j )| 2 on the set of all trigonometric polynomials of order at most n.To read more about function approximation by trigonometric polynomials see [1][2][3][4][5][6][7][8][9][10].
Also, in this paper we denote by c or c(b 1 , b 2 , . . ., b k ) some positive constants, which depend only on specified arguments (if any) and may vary from line to line, and by S n (f, x) the n-th partial sum of the Fourier series of the function f .We also note that it is easy to show that the Fourier series of any f ∈ C 2 Ω converges uniformly on R and the following representation is possible: where The goal of this work is to estimate the value |f Note that the special case of this problem is considered in [11], where the value In this work, we generalize the results from [11] for any function f ∈ C 2 Ω , as stated in the following theorem: Ω the following inequalities hold: The order of these estimates cannot be improved.
To prove this theorem we use a lemma from [12]:

Lemma 1 (Sharapudinov, [12]). If the Fourier series of f converges at the points
where holds true, where 2n < N and D n (x) is the Dirichlet kernel: This lemma considers only the case 2n < N .If 2n = N (when N is even) we can write (see [12]) where To prove the inequalities (3) and ( 4) from Theorem 1 we use the formulas where k ∈ N, α ∈ R, and Proof.Performing integration by parts two times we have From this we can get the estimate Lemma 3.For f ∈ C 2 Ω the following inequalities hold: Proof.Here we prove only (13) because the proof for inequality (14) can be found in [13, Theorem 2.1].Using (1) and (2) we can write 6) and (7)

. THE ESTIMATE FOR |R
n,N (f, x) are estimated later in this section, but first we prove three auxiliary lemmas.Lemma 4. For f ∈ C 0,1 Ω the following holds: Proof.Perform integration by parts: Repeat integration by parts for the last integral in (17): By moving the last integral from the right side to the left and dividing both sides by we get (16).
Proof.After performing Abel transformation (summation by parts) we have: Using Lemma 5 and the fact that Lemma 7. The following inequality holds: Proof.Using Lemma 2 we have Lemma 8.The following estimates hold: Proof.Rewrite (15) using (12): Using Corollary 1 we rewrite the above formula as follows: For brevity we only consider here estimation of R 2.1 n,N (f, x) R 2.2 n,N (f, x) can be estimated in almost the same way.Obviously, f ′ ∈ C 0,1 Ω , so we can apply Lemma 4 to R 2.1 n,N (f, x): Begin with R 2.1.2n,N (f, x).Applying Lemma 4 we get From this we can get the estimate In the same way we can get R 2.1.4n,N (f, x) c(f )/N 2 .Now we consider R 2.1.1 n,N (f, x) and R 2. 1.3  n,N (f, x) .We will estimate here only R 2.1.1 n,N (f, x) because the other one can be estimated in the similar way.After a simple transformation we have From this we have the uniform estimate for x ∈ R: Изв. Сарат.ун-та.Нов.сер.Сер.Математика.Механика.Информатика.2019.Т. 19, вып. 1 Using Lemma 6 and assuming α k = In the similar way we can get Finally, for R 2.1 n,N (f, x) we can write Using the same approach we can show that the value R 2.2 n,N (f, x) has the same estimate as R 2.1 n,N (f, x) , which leads us to ( 18) and ( 19).From the previous lemmas and inequality

. THE ESTIMATE FOR a
We can represent the above sum as S = S 1 + S 2 , where From the above formulas we can write the equation for S 1 − S 2 : Denote by G the set of numbers 22) by dividing it into two sums: For every k ∈ Ĝ we have |t 2k+1 −a i | > 2π/N (0 i m) so the points t 2k , t 2k+1 , t 2k+2 are inside some interval (a i , a i+1 ) and the function f ∈ C 2 Ω has an absolutely continuous derivative f ′′ on (a i , a i+1 ), therefore we can write also note that |G| 2m.Therefore, we have From ( 9) From ( 23) and (24) follows a

. THE PROOF OF THEOREM 1
The proof of Theorem 1 consists of two parts: first we prove that the inequalities (3) and ( 4) of the theorem hold, then we prove that these estimates cannot be improved for all f ∈ C 2 Ω .From the inequalities (10), (11), the estimates (13), ( 14), ( 20), ( 21) and Lemma 9 we can easily get (3) and (4).To prove that the order of these estimates cannot be improved we consider the aforementioned 2π-periodic function f Consider only the case when n < N/2.From (5) follows the inequality c(f )/N .Therefore, for every ε > 0 we can find a natural number N 0 , such that for every N > N 0 follows |R n,N (f, x)| < ε.Let N 0 (n) be a natural number such that for every N > N 0 (n) where E ⊂ R. So, we can write Lemma 10.The following inequalities hold: Proof.From [14, p. 443] we have the following representation: Hence we have R n f, π 4 c/n 2 .From (25) and the above lemma follows Theorem 1 is proved.

Lemma 5 . 2 . 2 . 6 .
The following estimate holds:Proof.The proof is obvious and follows from well-known formulas Lemma Let α 1 , α 2 , . . ., α n be a monotonous sequence (either increasing or decreasing) of n positive numbers.The following holds:

From
the previous equation we can get f (x) − S n (f, x) = − 4 1) 2 c/n.Now we consider the case when x = π/4 and n + 1 = 4l, l ∈ N. It is easy to show that